
The connection between mathematics and
art goes back thousands of years. Mathematics has been
used in the design of Gothic cathedrals, Rose windows,
oriental rugs, mosaics and tilings. Geometric forms were
fundamental to the cubists and many abstract expressionists,
and awardwinning sculptors have used topology as the
basis for their pieces. Dutch artist M.C. Escher represented
infinity, Möbius ands, tessellations, deformations,
reflections, Platonic solids, spirals, symmetry, and
the hyperbolic plane in his works.
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3DXplorMath
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Hilbert's SquareFilling Curve"Hilbert's SquareFilling Curve" by The
3DXM Consortium
In 1890 David Hilbert published a construction of a continuous curve whose image completely fills a square, which was a significant contribution to the understanding of continuity. Although it might be considered to be a pathological example, today, Hilbert's curve has become wellknown for a very different reasonevery computer science student learns about it because the algorithm has proved useful in image compression. See more fractal curves on the 3DXplorMath Gallery.
 adapted from "About Hilbert's Square Filling Curve" by Hermann Karcher


Mandelbrot SetA striking aspect of this image is its selfsimilarity: Parts of the set look very similar to larger parts of the set, or to the entire set itself. The boundary of the Mandelbrot Set is an example of a fractal, a name derived from the fact that the dimensions of such sets need not be integers like two or three, but can be fractions like 4/3. See more at the 3DXplorMath Fractal Gallery.
 Richard Palais (Univ. of California at Irvine, Irvine, CA)


Hopf Fibered Linked Tori"Hopf Fibered Linked Tori," by The
3DXM Consortium
The Hopf map maps the unit sphere in fourdimensional space to the unit sphere in threedimensional space. The four tori linked in this image are made up of fibers, or preimages, of the Hopf map. In this visualization, each fiber has a constant color and the color varies with the distance of the fibers. Any two of the four tori are linked, as are any pair of fibers on a given torus. See more surface images on the 3DXplorMath Gallery.
 adapted from "Hopf Fibration and Clifford Translation of the 3Sphere," by Hermann Karcher


Parametric Breather"Parametric Breather," by The 3DXM Consortium.
This striking object is an example of a surface in 3space whose intrinsic geometry is the hyperbolic geometry of Bolyai and Lobachevsky. Such surfaces are in onetoone correspondence with the solutions of a certain nonlinear waveequation (the socalled SineGordon Equation, or SGE) that also arises in highenergy physics. SGE is an equation of soliton type and the Breather surface corresponds to a timeperiodic 2soliton solution. See more pseudospherical surfaces on the 3DXplorMath Gallery.
 Richard Palais (Univ. of California at Irvine, Irvine, CA)





